Question

Find a quadratic polynomial, whose zeroes are 3 and -4​

Answer 1

Answer:

Step-by-step explanation:

Polynomials are types of expressions. And, an expression is a mathematical statement without an equal-to sign (=). Polynomial comprising a sum of terms, where each term holding a variable or variables is elevated to power and further multiplied by a coefficient. If α

and β

are zeroes of quadratic polynomial than the quadratic polynomial is given by x2−(α+β)x+αβ

where, (α+β)

is sum of zeroes and αβ

is product of zeroes.

We should know that, if x=α

and x=β

are the roots of the quadratic polynomial, then

(x−α)(x−β)

Solving this, we get,

x(x−β)−α(x−β)

Removing the brackets, we get,

x2−xβ−αx+αβ

⇒x2−x(β+α)+αβ

Rearranging this expression we get,

x2−(α+β)x+αβ

So, (x−α)(x−β)=x2−(α+β)x+αβ

where, α

and β

are zeroes of quadratic polynomial.

Given that,

α=−3⇒β=2

Now, we will find the values of sum of zeroes and product of zeroes.So,

Sum of zeroes =α+β

Substituting the values of α

and β

in the above expression, we get,

−3+2

Simplify this expression, we get −1

. Next, product of zeroes

αβ

Substituting the values of α

and β

in the above expression, we get,

(−3)(2)

Simplify this expression, we get −6

.

Thus, the quadratic polynomial we will as below,

x2−(α+β)x+αβ

Substituting the values of sum of the zeroes and the product of the zeroes, we will get,

x2−(−1)x+(−6)

Removing the brackets, we get,

x2+x−6

Hence, from the above explanation, we can say that the required polynomial we will,

(x−α)(x−β)=x2−(α+β)x+αβ

⇒(x−(−3))(x−2)=x2+x−6

Removing the brackets, we get,

∴(x+3)(x−2)=x2+x−6

Hence, from the above calculation, we can conclude that the quadratic polynomial having zeros −3

and 2

will be x2+x−6

.

Note: Here, the student should read the question carefully and understand what is given in the question and what is required to find out. Looking at the problem, it looks very easy, but still write the quadratic polynomial expression so that you don’t make any mistakes while solving the problem. There are three types of polynomials and they are – Monomials, Binomials and Trinomials. We are getting the answer which is trinomial because we have three terms here. We can verify the answer by drawing a graph.

Answer 2

Answer :-

Given :-

⟹ Zeroes of polynomial are 3 and -4.

To find :-

⟹ The quadratic polynomial.

Solution :-

Let's find,

》Sum of zeroes(α +β) = [3 +(-4)]

• -1

》Product of zeroes(αβ) = [3(-4)]

• -12

We know,

》Quadratic polynomial,

⇛ x² - (Sum of zeroes)x + (Product of zeroes)

⇛ x² - (-1)x + (-12)

⇛ x² + x - 12

∴,

x² + x - 12 is the required quadratic polynomial.